President-elect Donald Trump’s Twitter attack this week on a union official, followed by his choice of a labor secretary who has criticized new worker protections, has rattled leaders of the American labor movement, who fear unions may be facing their gravest crisis in decades.

On Thursday, Trump announced that he would nominate as his labor secretary Andrew Puzder, a fast-food executive who has opposed additional overtime pay for workers and expressed skepticism about increasing the minimum wage. That followed a pair of Twitter messages Wednesday evening in which Trump attacked an Indiana union leader who had criticized him, saying the official had done a “terrible job representing workers."...